Applied Partial Differential Equations

Front Cover......Page 1
Copyright......Page 4
Preface to the revised edition......Page 6
Preface to the first edition......Page 7
Contents......Page 8
Introduction......Page 14
1.1 Introduction......Page 19
1.2 Cauchy data......Page 21
1.3 Characteristics......Page 22
1.3.1 Linear and semilinear equations......Page 24
1.4 Domain of definition and blow-up......Page 26
1.5 Quasilinear equations......Page 28
1.6 Solutions with discontinuities......Page 32
* 1.7 Weak solutions......Page 35
* 1.8 More independent variables......Page 38
1.9 Postscript......Page 41
Exercises......Page 42
2.1 Motivation and models......Page 48
2.2 Cauchy data and characteristics......Page 54
2.3 The Cauchy-Kowalevski theorem......Page 58
2.4 Hyperbolicity......Page 61
2.4.1 Two-by-two systems......Page 62
2.4.2 Systems of dimension n......Page 63
2.4.3 Examples......Page 65
* 2.5 Weak solutions and shock waves......Page 68
2.5.1 Causality......Page 69
2.5.2 Viscosity and entropy......Page 72
2.5.3 Other discontinuities......Page 75
* 2.6 Systems with more than two independent variables......Page 76
Exercises......Page 81
3.1 Preamble......Page 89
3.2 The Cauchy problem for semilinear equations......Page 91
3.3 Characteristics......Page 93
3.4.1 Hyperbolic equations......Page 96
3.4.2 Elliptic equations......Page 97
3.4.3 Parabolic equations......Page 99
3.5 Some general remarks......Page 100
Exercises......Page 102
4.1 Introduction......Page 106
4.2.1 An ad hoc approach to Riemann functions......Page 107
4.2.2 The rationale for Riemann functions......Page 109
4.2.3 Implications of the Riemann function representation......Page 113
4.3 Non-Cauchy data for the wave equation......Page 115
* 4.3.1 Strongly discontinuous boundary data......Page 118
4.4 Transforms and eigenfunction expansions......Page 119
4.5.1 The wave equation in one space dimension......Page 126
4.5.2 Circular and spherical symmetry......Page 129
* 4.5.3 The telegraph equation......Page 131
* 4.5.5 General remarks......Page 132
4.6.1 The method of descent and Huygens\' principle......Page 133
4.6.2 Hyperbolicity and time-likeness......Page 138
4.7.1 Linear elasticity......Page 141
4.7.2 Maxwell\'s equations of electromagnetism......Page 144
4.8.1 Simple waves......Page 148
4.8.2 Hodograph methods......Page 150
4.8.3 Liouville\'s equation......Page 152
Exercises......Page 154
5.1.1 Gravitation......Page 164
5.1.2 Electromagnetism......Page 165
5.1.3 Heat transfer......Page 166
5.1.4 Mechanics......Page 168
5.1.5 Acoustics......Page 173
5.1.6 Aerofoil theory and fracture......Page 174
5.2.1 The Laplace and Poisson equations......Page 176
5.2.2 More general elliptic equations......Page 179
5.3 The maximum principle......Page 180
5.4 Variational principles......Page 181
5.5.1 The classical formulation......Page 182
5.5.2 Generalised function formulation......Page 184
5.6.1 Laplace\'s equation and Poisson\'s equation......Page 187
5.6.2 Helmholtz\' equation ......Page 193
5.6.3 The modified Helmholtz equation ......Page 195
5.7.1 Eigenvalues and eigenfunctions ......Page 196
5.7.2 Green\'s functions and transforms ......Page 197
5.8 Transform solutions of elliptic problems ......Page 199
5.8.1 Laplace\'s equation with cylindrical symmetry: Hankel transforms ......Page 200
5.8.2 Laplace\'s equation in a wedge geometry; the Mellin transform ......Page 203
* 5.8.3 Helmholtz\' equation ......Page 204
* 5.8.4 Higher-order problems ......Page 207
5.9 Complex variable methods ......Page 208
5.9.1 Conformal maps ......Page 210
* 5.9.2 Riemann-Hilbert problems ......Page 212
* 5.9.3 Mixed boundary value problems and singular integral equations ......Page 217
* 5.9.4 The Wiener-Hopf method ......Page 219
* 5.9.5 Singularities and index ......Page 222
*5.10 Localised boundary data ......Page 224
5.11.1 Nonlinear models ......Page 225
5.11.2 Existence and uniqueness ......Page 226
5.11.3 Parameter dependence and singular behaviour ......Page 228
5.12 Liouville\'s equation again ......Page 234
5.13 Postscript: V2 or -A ......Page 235
Exercises ......Page 236
6.1.1 Heat and mass transfer ......Page 254
6.1.2 Probability and finance ......Page 256
6.2 Initial and boundary conditions ......Page 258
6.3 Maximum principles and well-posedness ......Page 260
* 6.3.1 The strong maximum principle ......Page 261
6.4.1 Green\'s functions: general remarks ......Page 262
6.4.2 The Green\'s function for the heat equation with no boundaries ......Page 264
6.4.3 Boundary value problems ......Page 267
* 6.4.4 Convection-diffusion problems ......Page 273
6.5 Similarity solutions and groups ......Page 275
6.5.1 Ordinary differential equations ......Page 277
6.5.2 Partial differential equations ......Page 278
* 6.5.3 General remarks......Page 282
6.6.1 Models......Page 284
6.6.3 Similarity solutions and travelling waves......Page 288
6.6.4 Comparison methods and the maximum principle......Page 294
* 6.6.5 Blow-up......Page 297
* 6.7 Higher-order equations and systems......Page 299
6.7.1 Higher-order scalar problems......Page 300
6.7.2 Higher-order systems......Page 302
Exercises......Page 304
7.1 Introduction and models......Page 318
7.1.1 Stefan and related problems......Page 319
7.1.2 Other free boundary problems in diffusion......Page 323
7.1.3 Some other problems from mechanics......Page 327
7.2 Stability and well-posedness......Page 331
7.2.1 Surface gravity waves......Page 332
7.2.2 Vortex sheets......Page 334
7.2.3 Hele-Shaw flow......Page 335
7.2.4 Shock waves......Page 337
7.3.1 Comparison methods......Page 339
7.3.2 Energy methods and conserved quantities......Page 340
7.3.3 Green\'s functions and integral equations......Page 341
* 7.4 Weak and variational methods......Page 342
7.4.1 Variational methods......Page 343
7.4.2 The enthalpy method......Page 348
7.5 Explicit solutions......Page 351
7.5.1 Similarity solutions......Page 352
* 7.5.2 Complex variable methods......Page 354
* 7.6 Regularisation......Page 358
* 7.7 Postscript......Page 360
Exercises......Page 362
8.1 Introduction......Page 372
8.2.1 Two independent variables......Page 373
8.2.3 The eikonal equation......Page 379
* 8.2.4 Eigenvalue problems......Page 387
8.2.5 Dispersion......Page 389
8.2.6 Bicharacteristics......Page 390
*8.3 Hamilton-Jacobi equations and quantum mechanics......Page 391
*8.4 Higher-order equations......Page 393
Exercises......Page 396
9.1 Introduction......Page 406
9.2 Linear systems revisited......Page 408
9.2.1 Linear systems: Green\'s functions......Page 409
9.2.2 Linear elasticity......Page 411
9.2.3 Linear inviscid hydrodynamics......Page 413
9.2.4 Wave propagation and radiation conditions......Page 416
9.3 Complex characteristics and classification by type......Page 418
9.4.1 Heat conduction with ohmic heating......Page 420
9.4.2 Space charge......Page 421
9.4.4 Inviscid flow: the Euler equations......Page 422
9.4.5 Viscous flow......Page 425
9.5.1 Fluid/solid acoustic interactions......Page 427
9.6 Gauges and invariance......Page 428
9.7 Solitons......Page 430
Exercises......Page 439
Conclusion......Page 447
References......Page 449
Index......Page 452
Back Cover......Page 464